Step-by-step Solution

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Step-by-step explanation

Problem to solve:

$\lim_{x\to0}\left(\frac{5^x-1}{\ln\left(1+x\right)}\right)$

Learn how to solve limits problems step by step online.

$\lim_{x\to0}\left(\frac{\frac{d}{dx}\left(5^x-1\right)}{\frac{d}{dx}\left(\ln\left(1+x\right)\right)}\right)$

Learn how to solve limits problems step by step online. Evaluate the limit of (5^x-1)/(ln(1+x) as x approaches 0. If we try to evaluate the limit directly, it results in indeterminate form. Then we need to apply L'Hôpital's rule. The derivative of the natural logarithm of a function is equal to the derivative of the function divided by that function. If f(x)=ln\:a (where a is a function of x), then \displaystyle f'(x)=\frac{a'}{a}. The derivative of a sum of two functions is the sum of the derivatives of each function. The derivative of the constant function (-1) is equal to zero.

$\ln\left(5\right)$$\,\,\left(\approx 1.6094379124341003\right)$
$\lim_{x\to0}\left(\frac{5^x-1}{\ln\left(1+x\right)}\right)$